Concepts

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Updated2025-11-25
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The FFT of an image is a 2D array of complex numbers, also represented as a complex image. It represents the frequencies of occurrence of light-intensity variations in the spatial domain. The low frequencies correspond to smooth and gradual intensity variations found in the overall patterns of the source image. The high frequencies correspond to abrupt and short intensity variations found at the edges of objects, around noisy pixels, and around details.

FFT Representation

There are two possible representations of the Fast Fourier transform of an image: the standard representation and the optical representation.

Standard Representation

In the standard representation, high frequencies are grouped at the center of the image while low frequencies are located at the edges. The constant term, or null frequency, is in the upper-left corner of the image.

The frequency range is as follows:

(- \frac{N}{2}, \frac{N}{2}) \times (- \frac{M}{2}, \frac{M}{2})

where M is the horizontal resolution of the image, and N is the vertical resolution of the image.

Note Vision uses the standard representation to represent complex images in memory. Use this representation when building an image mask.

The illustration shows an original image:

The illustration shows the FFT of the same image using standard representation:

Optical Representation

In the optical representation, low frequencies are grouped at the center of the image while high frequencies are located at the edges. The constant term, or null frequency, is at the center of the image.

The frequency range is as follows:

(- \frac{N}{2}, \frac{N}{2}) \times (- \frac{M}{2}, \frac{M}{2})

The illustration shows an original image:

The illustration shows the FFT of the same image using optical representation:

Note Vision uses optical representation when displaying a complex image.

You can switch from standard representation to optical representation by permuting the A, B, C, and D quarters.

Intensities in the FFT image are proportional to the amplitude of the displayed component.

Lowpass FFT Filters

A lowpass frequency filter attenuates, or removes, high frequencies present in the FFT plane. This filter suppresses information related to rapid variations of light intensities in the spatial image. In this case, an inverse FFT produces an image in which noise, details, texture, and sharp edges are smoothed.

A lowpass frequency filter attenuates, or removes, spatial frequencies located outside a frequency range centered on the fundamental (or null) frequency.

Lowpass Attenuation

Lowpass attenuation applies a linear attenuation to the full frequency range, increasing from the null frequency f₀ to the maximum frequency f_max. This is done by multiplying each frequency by a coefficient C, which is a function of its deviation from the fundamental and maximum frequencies.

C_{f} = \frac{f m a x - f}{f_{m a x} - f_{0}}

where,

$C (f_{0}) = 1$
$C (f_{m a x}) = 0$

Lowpass Truncation

Lowpass truncation removes a frequency f if it is higher than the cutoff or truncation frequency, f_c. This is done by multiplying each frequency f by a coefficient C equal to 0 or 1, depending on whether the frequency f is greater than the truncation frequency f_c.

If f > f_c then C(f) = 0 else C(f) = 1

The following series of graphics illustrates the behavior of both types of lowpass filters. They represent the 3D-view profile of the magnitude of the FFT.

This example uses the following original FFT.

After lowpass attenuation, the magnitude of the central peak is the same, and variations at the edges almost have disappeared.

After lowpass truncation with f_c = f₀ + 20%( f_max – f₀), spatial frequencies outside the truncation range [ f₀, f_c] are removed. The part of the central peak that remains is identical to the one in the original FFT plane.

Highpass FFT Filters

A highpass FFT filter attenuates, or removes, low frequencies present in the FFT plane. It has the effect of suppressing information related to slow variations of light intensities in the spatial image. In this case, the Inverse FFT command produces an image in which overall patterns are attenuated and details are emphasized.

Highpass Attenuation

Highpass attenuation applies a linear attenuation to the full frequency range, increasing from the maximum frequency f_max to the null frequency f₀. This is done by multiplying each frequency by a coefficient C, which is a function of its deviation from the fundamental and maximum frequencies.

where,

$C (f_{0}) = 0$
$C (f_{m a x}) = 1$

Highpass Truncation

Highpass truncation removes a frequency f if it is lower than the cutoff or truncation frequency, f_c. This is done by multiplying each frequency f by a coefficient C equal to 1 or 0, depending on whether the frequency f is greater than the truncation frequency f_c.

I f f < f_{c}

T h e n C_{f} = 0

E l s e C_{f} = 1

The following series of graphics illustrates the behavior of both types of highpass filters. They represent the 3D-view profile of the magnitude of the FFT. This example uses the following original FFT image.

After highpass attenuation, the central peak has been removed, and variations present at the edges remain.

After highpass truncation with f_c = f₀ + 20%( f_max – f₀), spatial frequencies inside the truncation range [f₀, f_c] are set to 0. The remaining frequencies are identical to the ones in the original FFT plane.

Mask FFT Filters

A mask FFT filter removes frequencies contained in a mask specified by the user. Depending on the mask definition, this filter can act as a lowpass, bandpass, highpass, or any type of selective filter.

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